How does antimatter come into 'existence' within a particle accelerator?

Particles moving very quickly (near the speed of light) can convert their energy of motion into rest mass energy of new particles when they collide with other particles. So for instance a proton accelerated to high speeds when it rams into another proton can produce a shower of particles whose total mass is much more than the rest mass of the two protons you start with. This is the conversion of energy into mass, the reverse process of turning mass into energy, which occurs in a Nuclear bomb or reactor.

When these new particles are made there are roughly as many matter as anti-matter particles produced although the details depends on the nature of the two particles in the collision, the energy of the collision etc etc. The point is that matter and anti-matter are made equally in the process.

We do think there is a slight asymmetry in this process that leads to the Universe being made mainly of matter rather than anti-matter, but this asymmetry is very slight.

We do think there is a slight asymmetry in this process that leads to the Universe being made mainly of matter rather than anti-matter, but this asymmetry is very slight.

Is this asymmetry small enough that some of the visible universe is still composed of anti-matter? Or, conversely, has all anti-matter in the visible universe been annihilated and everything we see is just the asymmetry?

The second option. We are pretty sure that there are no accumulations of anti-matter anywhere (such as say an anti-matter planet, or even as small as an anti-matter grain of dust). As you suggest, we think that when the Universe was first cool enough to form atomic particles, the matter and anti-matter produced annihilated each other in equal amounts leaving only a small amount of matter due to the asymmetry.